Basic Topology 3: Algebraic Topology and Topology of Fiber Bundles

This third of the three-volume book is targeted as a basic course in algebraic topology and topology for fiber bundles for undergraduate and graduate students of mathematics. It focuses on many variants of topology and its applications in modern analysis, geometry, and algebra. Topics covered in this volume include homotopy theory, homology and cohomology theories, homotopy theory of fiber bundles, Euler characteristic, and the Betti number. It also includes certain classic problems such as the Jordan curve theorem along with the discussions on higher homotopy groups and establishes links between homotopy and homology theories, axiomatic approach to homology and cohomology as inaugurated by Eilenberg and Steenrod. It includes more material than is comfortably covered by beginner students in a one-semester course. Students of advanced courses will also find the book useful. This book will promote the scope, power and active learning of the subject, all the while covering a wide range of theory and applications in a balanced unified way.

這本三卷書的第三卷針對數學本科和研究生的學生，作為代數拓撲學和纖維束拓撲學的基礎課程。它聚焦於拓撲學的多種變體及其在現代分析、幾何和代數中的應用。本卷涵蓋的主題包括同伦理论、同调和上同调理论、纖維束的同伦理论、欧拉特征和贝蒂数。它還包括一些經典問題，如Jordan曲線定理，以及對高阶同伦群的討論，並建立同伦理论和同调理论之間的联系，以及Eilenberg和Steenrod所開創的公理化同调和上同调方法。本書包含的內容超出了一個學期課程中初學者學生的舒適範圍。高級課程的學生也會發現這本書很有用。本書將促進這一學科的範圍、能力和積極學習，同時以平衡統一的方式涵蓋了廣泛的理論和應用。

MAHIMA RANJAN ADHIKARI, Ph.D., M.Sc. (Gold Medalist), is the founder president of the Institute for Mathematics, Bioinformatics and Computer Science (IMBIC), Kolkata, India. He is a former professor at the Department of Pure Mathematics, University of Calcutta, India. His research papers are published in national and international journals of repute, including the Proceedings of American Mathematical Society. He has authored nine textbooks and is the editor of two, including: Basic Modern Algebra with Applications (Springer, 2014), Basic Algebraic Topology and Applications (Springer, 2016), and Mathematical and Statistical Applications in Life Sciences and Engineering (Springer, 2017).

Twelve students have been awarded Ph.D. degree under his guidance on various topics such as algebra, algebraic topology, category theory, geometry, analysis, graph theory, knot theory and history of mathematics. He has visited several universities and research institutions in India, USA, UK, Japan, China, Greece, Sweden, Switzerland, Italy, and many other counties on invitation. A member of the American Mathematical Society, Prof. Adhikari is on the editorial board of several journals of repute. He was elected as the president of the Mathematical Sciences Section (including Statistics) of the 95th Indian Science Congress, 2008. He has successfully completed research projects funded by the Government of India.

MAHIMA RANJAN ADHIKARI博士，M.Sc.（金牌得主），是印度加爾各答數學、生物信息學和計算機科學研究所（IMBIC）的創始主席。他曾是印度加爾各答大學純數學系的教授。他的研究論文發表在國內外有聲望的期刊上，包括《美國數學學會會議記錄》。他撰寫了九本教科書，並擔任兩本書的編輯，包括：《基礎現代代數及其應用》（Springer，2014年），《基礎代數拓撲及其應用》（Springer，2016年）和《生命科學和工程中的數學和統計應用》（Springer，2017年）。

在他的指導下，有十二名學生在代數、代數拓撲、范畴論、幾何、分析、圖論、結糾論和數學歷史等各個領域獲得了博士學位。他曾應邀訪問印度、美國、英國、日本、中國、希臘、瑞典、瑞士、意大利和其他許多國家的多所大學和研究機構。作為美國數學學會的成員，Adhikari教授是多個有聲望期刊的編輯委員會成員。他曾當選為第95屆印度科學大會（2008年）數學科學部（包括統計學）的主席。他成功完成了由印度政府資助的研究項目。